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Full text of "Treatise On Analysis Vol-Ii"

168       XIII    INTEGRATION

mapping gmn: X-> [0,1] such that gmn(x) = 1 for all x e Um and gmn(x) = 0 for all
jc e X — Un; apply Problem 10(e) to each of the functions gmn. Finally, to show that
E has measure 1, show (with the notation of Problem 10) that, for any measure v el,

•)(x))2 d[jLx(y) = 0

for almost all x e X and all/e <*f(X), by applying Problem 10(d); now let/run through
a dense sequence of functions in ^(X).)

(b)    With respect to the vague topology on M(X) show that, for every measure v e I>

ixdv(x)

in the sense of (13.10), and that the external points of I are the measures which are
ergodic with respect to u (cf. Section 13.10, Problem 9).

(c)    For  every   ergodic  measure  v e I,   let   Qv (resp. Rv)   be   the set  of points
x e Q (resp. x e R) such that px = v. The measure v is concentrated on Rv (13.18). The
sets Qv and Rv are called the quasi-ergodic set and the ergodic set corresponding to v.
The Qv (resp. Rv) form a partition of Q (resp. R), and we have w(Qv) = Qv and
w(Rv) = Rv. Show that, for each closed set F such that w(F) = F, we have either
F n Rv = Rv or F n Ry= 0.

(d)    For every nonempty closed set F such that w(F) = F, show that R n F ^ 0
(consider the points which are regular with respect to u \ F).

(e)    Let Jt be the monoid generated by lx and u. Show that if Z is any minimal
closed orbit with respect to ^ (Section 12.10, Problem 6) we have u(Z) = Z and hence
R n Z 9*= 0. Deduce that, for each x e X, the closed orbit O(x) of x with respect to
^ intersects R, and that O(x) n R is a union of ergodic sets Rv.

(f)    For each x e X, let jit be a measure which is a cluster point of the sequence (pn, x)
in M(X). Show that ft(O(jc)) = 1.

(g)    Show that for a measure (JL e I to be ergodic it is necessary and sufficient that
there should exist an ergodic set Rv such that /x(Rv) = 1, and that then we have ju. = v.
(Use (b) to prove that the condition is sufficient.)

(h) If A; e X is such that O(x) contains only one ergodic set, show that x is an ergodic
point (use (f) and (g)).

(i) Suppose that I consists of a single measure v0. Then Q = X, and R = D is the
support of VQ (use (h)) and is the only minimal closed orbit with respect to M. More-
over, for each function /e ^(X), the sequence of numbers - ]T /("*(*)) converges to

n k=o

\fdvQ uniformly in x e X. (Observe that, for each n, the mapping xt-+pn> x of X into
M(X) is continuous with respect to the vague topology, and use (7.5.6).)

12.   For every real number p such that 0 < p < +00, and every mapping /: X ->R, put

(a)    If p > 1, put q = p/(p — 1). If/, g are any two mappings of X into R, show that
NiCfc) ^ NP(/)N,(0) ("Holder's inequality "). (Show that the set of points fo, t2) e R2
such that fi ^ 0 and r2 ^ 0 and t\lp tŁ/g ^ 1 is convex, and argue as in Section 13.8,
Problem 14(c), making use of (13.5.6). Notice that the proof of (13.11.2.2) is a particular
case.)

(b)    If / > 0, g > 0 and p > 1, then Np(/ + #) < Np(/) + Np(#) (" Minkowski's in-
equality"). (Same method as (a): consider the set of points (tt, t2) such that rx > 0
and t2 > 0 and t\tp + t\'p > 1.)eparable real Frechet space, K a compact convex subset of E; let u be a